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Tunable Orbits Influence in a Driven Stadium-Like Billiard

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A Mathematical Modeling Approach from Nonlinear Dynamics to Complex Systems

Part of the book series: Nonlinear Systems and Complexity ((NSCH,volume 22))

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Abstract

The dynamics of a driven stadium-like billiard is investigated through a four-dimensional nonlinear mapping. We set a critical resonance velocity, which plays the role of an ensemble separation according to the initial velocities. When the resonance is active, the invariant curves that surround the stability islands become stochastic layers, thus allowing a change in the dynamics from chaos to stability and vice versa, leading the root mean square velocity to steady state plateaus for long times. A tunneling behavior of orbits in the lower ensemble was characterized via transport analysis and Lyapunov exponents. Our results may be extended to other similar dynamical systems that may present similar critical resonances.

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Acknowledgements

ALPL acknowledges FAPESP (2014/25316-3) and FAPESP (2015/26699-6) for financial support. ALPL also thanks the University of Bristol for the kind hospitality during his stay in UK. This research was supported by resources supplied by the Center for Scientific Computing (NCC/GridUNESP) of the São Paulo State University (UNESP). The author also acknowledges Alexander Loskutov (in memorian) for the art of Fig. 4.1.

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Authors and Affiliations

  1. Departamento de Física, UNESP - Univ Estadual Paulista, Rio Claro, SP, Brazil

    André L. P. Livorati

  2. School of Mathematics, University of Bristol, Bristol, UK

    André L. P. Livorati

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  1. André L. P. Livorati
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Correspondence to André L. P. Livorati .

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  1. Instituto Nacional de Pesquisas Espaciais – INPE, São José dos Campos, Brazil

    Elbert E. N. Macau

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Livorati, A.L.P. (2019). Tunable Orbits Influence in a Driven Stadium-Like Billiard. In: Macau, E. (eds) A Mathematical Modeling Approach from Nonlinear Dynamics to Complex Systems . Nonlinear Systems and Complexity, vol 22. Springer, Cham. https://doi.org/10.1007/978-3-319-78512-7_4

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  • DOI: https://doi.org/10.1007/978-3-319-78512-7_4

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